Decomposition and framing of F-bundles and applications to quantum cohomology

TL;DR

This paper develops a spectral decomposition theorem for F-bundles, a non-archimedean analogue of variation of nc-Hodge structures, with applications to quantum cohomology and birational invariants.

Contribution

It establishes the spectral decomposition theorem for F-bundles and proves the uniqueness of the decomposition map in quantum cohomology contexts.

Findings

Spectral decomposition theorem for F-bundles established

Uniqueness of the decomposition map for quantum cohomology confirmed

Extension of framing for logarithmic F-bundles proven

Abstract

F-bundle is a formal/non-archimedean version of variation of nc-Hodge structures which plays a crucial role in the theory of atoms as birational invariants from Gromov-Witten theory. In this paper, we establish the spectral decomposition theorem for F-bundles according to the generalized eigenspaces of the Euler vector field action. The proof relies on solving systems of partial differential equations recursively in terms of power series, and on estimating the size of the coefficients for non-archimedean convergence. The same technique allows us to establish the existence and uniqueness of the extension of framing for logarithmic F-bundles. As an application, we prove the uniqueness of the decomposition map for the A-model F-bundle (hence quantum D-module and quantum cohomology) associated to a projective bundle, as well as to a blowup of an algebraic variety. This complements the existence results by Iritani-Koto and Iritani.